2022/09/27 by Ville Salo, Salo, Ville
Computer Science · Materials Science · #Cellular Automata and Applications #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #Quasicrystal Structures and Properties #Rings and Algebras (math.RA) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.2209.13373
openalex publication_date 2022/09/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We show that a cellular automaton on a one-dimensional two-sided mixing subshift of finite type is a von Neumann regular element in the semigroup of cellular automata if and only if it is split epic onto its image in the category of sofic shifts and block maps. It follows from previous joint work of the author and Törmä that von Neumann regularity is a decidable condition, and we decide it for all elementary CA, obtaining the optimal radii for weak generalized inverses. Two sufficient conditions for non-regularity are having a proper sofic image or having a point in the image with no preimage of the same period. We show that the non-regular ECA 9 and 28 cannot be proven non-regular using these methods. We also show that a random cellular automaton is non-regular with high probability.